An alternative functional equation of Jensen type on groups

Choodech Srisawat, Nataphan Kitisin*, Paisan Nakmahachalasint

ABSTRACT: Given an integer λ≠2, we establish the general solution of an alternative functional equation of Jensen type on certain groups. First, we give a criterion for the existence of the general solution for the functional equation f(xy−1)−2f(x)+f(xy)=0 or f(xy−1)−λf(x)+f(xy)=0, where f is a mapping from a group (G,⋅) to a uniquely divisible abelian group (H,+). Then we show that, for λ∉{0,−1,−2}, the above alternative functional equation is equivalent to the classical Jensen's functional equation. We also find the general solution in the case when G is a cyclic group and λ≠2 is an integer.